Simplify the following expression and state the condition under which the simplification is valid. $k = \dfrac{-10x^2 - 30x + 100}{x^3 + 7x^2 - 18x}$
Explanation: First factor out the greatest common factors in the numerator and in the denominator. $ k = \dfrac {-10(x^2 + 3x - 10)} {x(x^2 + 7x - 18)} $ $ k = -\dfrac{10}{x} \cdot \dfrac{x^2 + 3x - 10}{x^2 + 7x - 18} $ Next factor the numerator and denominator. $ k = - \dfrac{10}{x} \cdot \dfrac{(x - 2)(x + 5)}{(x - 2)(x + 9)}$ Assuming $x \neq 2$ , we can cancel the $x - 2$ $ k = - \dfrac{10}{x} \cdot \dfrac{x + 5}{x + 9}$ Therefore: $ k = \dfrac{ -10(x + 5)}{ x(x + 9)}$, $x \neq 2$